Explicit 3-colorings for exponential graphs
Adrien Argento, Pierre Charbit, Alantha Newman

TL;DR
This paper presents an explicit polynomial-time algorithm for 3-coloring exponential graphs, providing a constructive method that affirms a conjecture about the chromatic properties of these graphs and their relation to graph products.
Contribution
It introduces a new explicit polynomial-time algorithm for 3-coloring exponential graphs, answering a previously open question and offering an alternative proof for a known chromatic property.
Findings
The algorithm efficiently finds 3-colorings in exponential graphs.
It confirms that the categorical product of two 4-chromatic graphs is 4-chromatic.
Provides a constructive approach to a problem previously solved only implicitly.
Abstract
For a graph and integer , two functions from into are adjacent if for all edges of , . The graph of all such functions is the exponential graph . El-Zahar and Sauer proved that if , then is 3-chromatic. Tardif showed that, implicit in their proof, is an algorithm for 3-coloring whose time complexity is polynomial in the size of . Tardif then asked if there is an "explicit" algorithm for finding such a coloring: Essentially, given a function belonging to a 3-chromatic component of , can we assign a color to this vertex in time polynomial in the size of ? The main result of this paper is to present such an algorithm, answering Tardif's question affirmatively. Our algorithm yields an alternative proof of the theorem of El-Zahar and Sauer that the categorical…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Limits and Structures in Graph Theory
